AQA A Level Mathematics: Overarching Themes — Practice Questions

These are original questions written for Marlbridge, in the style and at the standard of the examination. They are not reproduced past-paper questions — examination boards hold copyright in their own papers. Use these alongside the official past papers available free from your board.

Related: Overarching Themes revision notes

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Section A

1. Explain the difference between the symbols ⇒, ⇐ and ⇔, giving an example of each. [3]

2. Explain the difference between a conjecture, a counter-example and a proof. [3]

Section B

3. Consider the statement: “If n is a prime number then n is odd.”

(a) Give a counter-example. [1] (b) State whether the converse is true and justify your answer. [2]

4. Prove by exhaustion that no square number ends in 2, 3, 7 or 8. [4]

5. Prove by deduction that the product of two consecutive even numbers is a multiple of 8. [4]

6. A model for the height of a ball is h = 20t − 5t².

(a) State two assumptions made in this model. [2] (b) Explain the effect on the model of including air resistance. [2] (c) Explain why a model may still be useful even though its assumptions are not exactly true. [2]

7. Explain what is meant by “necessary and sufficient”, using the statement “a quadrilateral is a square” and “a quadrilateral has four equal sides”. [3]

8. Outline the structure of a proof by contradiction, and name one classic result usually proved this way. [3]

9. Explain the difference between a question that says “hence” and one that says “hence or otherwise”. [2]

10. State two reasons why rounding partway through a calculation, rather than at the end, can lose marks. [2]

11. State one way a calculator can be misused in an exam question that requires working to be shown. [2]

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Answers

1. ⇒ means “implies”: the first statement leads to the second, e.g. x = 3 ⇒ x² = 9 [1]. ⇐ means “is implied by”, so the second leads to the first [1]. ⇔ means “if and only if”, so each implies the other, e.g. a triangle is equilateral ⇔ all three angles are 60° [1].

2. A conjecture is a statement believed to be true but not yet proved [1]. A counter-example is a single case that shows a statement is false [1]. A proof is a logical argument establishing that a statement is true in every case [1].

3. (a) n = 2, which is prime but even [1]. (b) The converse is “if n is odd then n is prime” — this is false [1]; 9 is odd but not prime [1].

4. Every integer ends in one of the digits 0 to 9 [1]. Squaring each in turn gives final digits 0, 1, 4, 9, 6, 5, 6, 9, 4, 1 [1] [1]. None of these is 2, 3, 7 or 8, and the final digit of a square depends only on the final digit of the number, so the statement holds for all integers [1].

5. Let the numbers be 2n and 2n + 2 [1]. Their product is 4n(n + 1) [1]. Of any two consecutive integers n and n + 1, one must be even, so n(n + 1) is a multiple of 2 [1]. Hence 4n(n + 1) is a multiple of 8 [1].

6. (a) Any two: air resistance is negligible [1]; the ball is a point mass; g is constant at 10 m s⁻²; the ball is thrown from ground level [1]. (b) The ball would not rise as high and would not be symmetric about the highest point [1]; the descent would take longer than the ascent, so the simple quadratic would no longer fit [1]. (c) A model simplifies reality to make the mathematics tractable [1]; if the predictions are close enough for the purpose in hand, the simplification is justified — and the model can be refined later if greater accuracy is needed [1].

7. Having four equal sides is a necessary condition for a square — every square has them [1] — but it is not sufficient, because a rhombus also has four equal sides without being a square [1]. A necessary and sufficient condition would be four equal sides and four right angles [1].

8. Assume the opposite of the statement is true, derive a logical contradiction from that assumption, and conclude that the original statement must be true [2]. A classic example is the proof that √2 is irrational (or the infinitude of primes) [1].

9. “Hence” requires the answer to be built from the result of the previous part [1]. “Hence or otherwise” allows a completely fresh method, not derived from the earlier part, provided it reaches the correct result [1].

10. Rounding partway through introduces error that compounds through later steps, so the final answer may be inaccurate even if the method is correct [1]; it also risks the final answer being judged wrong to the required accuracy, since exam mark schemes expect full accuracy carried through and only rounded at the very end [1].

11. Giving an unsupported answer straight from the calculator, with no working shown, earns no method marks where working is explicitly required — the calculator should be used to check a method, not replace it [2].

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Where marks are usually lost

• Confusing a statement with its converse.
• Giving examples rather than a general argument in a deductive proof.
• Listing assumptions that the model does not actually make.
• Saying “necessary” when the condition is in fact sufficient.